Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 90742, 22 pages

The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications

Zehua Zhou and Yan Liu

Department of Mathematics, Tianjin University, Tianjin 300072, China

Received 27 December 2005; Revised 26 June 2006; Accepted 22 July 2006

Copyright © 2006 Zehua Zhou and Yan Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let Un be the unit polydisc of n and φ=(φ1,,φn) a holomorphic self-map of Un. p(Un), 0p(Un), and B0*p(Un) denote the p-Bloch space, little p-Bloch space, and little star p-Bloch space in the unit polydisc Un, respectively, where p,q>0. This paper gives the estimates of the essential norms of bounded composition operators Cφ induced by φ between p(Un) (0p(Un) or B0*p(Un)) and q(Un) (0q(Un) or B0*p(Un)). As their applications, some necessary and sufficient conditions for the (bounded) composition operators Cφ to be compact from p(Un) (0p(Un) or 0p(Un)) into q(Un) (0q(Un) or B0*p(Un)) are obtained.